Lecture 5

Outline

• Why spatial modelling?
• Why is spatial modelling computationally expensive?
• Different data types?
• Modelling in discrete space – areal data
• Modelling in continuous space – geo-referenced data
• Modelling continuous space – spatial point process data

Spatial modelling

many natural processes take place in space large amounts of data collected in space; increased resolution large, complex data sets

HOWEVER:

• spatial statistical analysis is often not complex enough
• inaccessible to practitioners as literature written for statisticians
• development of methodology often not linked to applications (unrealistic assumptions)
• difficult to apply (unless you are an expert statistician and programmer)

we will see that inlabru can help with this… why do we need spatial models in the first place?

an example

global pm 2.5

exposure to air pollution; particulate matter < 2.5 microns in diameter (PM 2.5)

• linked to poor health outcomes
• responsible for three million deaths worldwide each year
• maybe observations not independent?
• sparsely measured
• heterogeneous spatial coverage

We need to take account of spatial dependence… i.e. account for autocorrelation… and complexity

Spatial modelling

accounting for spatial dependence

• standard statistical modelling usually assumes independent observations

• distributional assumptions that are made are only true if the independence assumption hold

• spatio-temporal data, however, are often not independent, but are spatially auto correlated

• independence assumptions are violated here

• two observations taken in close proximity are very similar

• do not provide two (independent) pieces of information

Spatial modelling

accounting for spatial dependence - ignoring this = pretend we have as much independent information as we have observations - pretend we have more information than we actually have - spurious inference and ultimately wrong conclusions…

spatio-temporal models have special model components that explicitly model the dependence structure we need to:

• say what the dependence in our data looks like in general: choose a specific class of spatial models, and
• estimate its specific properties for a specific dataset

Spatial modelling – computations

• computationally expensive
• in the past: often MCMC, takes forever

Types of spatial data

We can distinguish three types of spatial data

Discrete space: - data on a spatial grid (areal data)

Continuous space: - geostatistical (geo-referenced) data - spatial point data

Discrete space: areal data

• data on a (regular or irregular) spatial grid

• examples: number of individuals in a region, average rainfall in a province

• (originally geostatistical or point data; gridded for practical reasons)

Observed response(s): Measurement over each grid cell (e.g. number of individuals in cell; rainfall in province)

Continuous space: geostatistical data

• phenomenon that is continuous in space
• examples: nutrient levels in soil, salinity in the sea measurements at a given set of locations that are determined by surveyor

Observed response(s): measurement(s) taken at given locations

Continuous space: spatial point patterns

• patterns formed by locations of objects (individuals) in space (typically 2D)
• examples: locations of trees in a forest, groups of animals, earthquakes

Observed response(s): x,y coordinates of points (individuals/groups) sometimes also properties of individuals/groups (“marks”)

point patterns vs. geostatistical data

point patterns: - data format : x,y coordinates optional : properties of objects represented by the points (“marks”)

geostatistical data: - data format : x,y coordinates not optional : measurement taken in these locations

These seem rather similar…

point patterns vs. geostatistical data

point patterns : - data format : x,y coordinates

optional : properties of objects represented by the points (“marks”)

• aim : modelling the locations of objects in continuous space

• locations are being modelled and are considered random

• marks only take values in locations where there is a point

geostatistical data : - data format : x,y coordinates

not optional : measurement taken in these locations

• aim : modelling continuous process observed in finite number of locations

• locations have typically been deliberately chosen and are fixed

• continuous spatial field takes on values in whole subset of \(\mathbf R^ 2\)

Take home message!

• all spatial models we discuss here are also special cases of the large class of Latent Gaussian models

• to account for spatial dependence in the data, different types of spatial terms have to be included in the model for different spatial data structures, but they are all approximated by an SPDE model

• inlabru provides an efficient and unified way to fit all these models!